On the existence and structure of a mush at the inner
core boundary of the Earth
Renaud Deguen, Thierry Alboussière, Daniel Brito
To cite this version:
Renaud Deguen, Thierry Alboussière, Daniel Brito. On the existence and structure of a mush
at the inner core boundary of the Earth. Physics of the Earth and Planetary Interiors, Elsevier,
2007, 164 (1-2), pp.36-49. .
HAL Id: hal-00207822
https://hal.archives-ouvertes.fr/hal-00207822
Submitted on 18 Jan 2008
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On the existence and structure of a mush
at the inner core boundary of the Earth
R. Deguen ∗ , T. Alboussière, D. Brito
Laboratoire de Géophysique Interne et Tectonophysique, Université Joseph
Fourier, Grenoble, France
hal-00207822, version 1 - 18 Jan 2008
Abstract
It has been suggested about 20 years ago that the liquid close to the inner core
boundary (ICB) is supercooled and that a sizable mushy layer has developed during
the growth of the inner core. The morphological instability of the liquid-solid interface which usually results in the formation of a mushy zone has been intensively
studied in metallurgy, but the freezing of the inner core occurs in very unusual conditions: the growth rate is very small, and the pressure gradient has a key role, the
newly formed solid being hotter than the adjacent liquid.
We investigate the linear stability of a solidification front under such conditions,
pointing out the destabilizing role of the thermal and solutal fields, and the stabilizing role of the pressure gradient. The main consequence of the very small solidification rate is the importance of advective transport of solute in liquid, which tends
to remove light solute from the vicinity of the ICB and to suppress supercooling,
thus acting against the destabilization of the solidification front. For plausible phase
diagrams of the core mixture, we nevertheless found that the ICB is likely to be
morphologically unstable, and that a mushy zone might have developed at the ICB.
The thermodynamic thickness of the resulting mushy zone can be significant, from
∼ 100 km to the entire inner core radius, depending on the phase diagram of the
core mixture. However, such a thick mushy zone is predicted to collapse under its
own weight, on a much smaller length scale (. 1 km). We estimate that the interdendritic spacing is probably smaller than a few tens of meter, and possibly only a
few meters.
Key words: inner core boundary, morphological instability, mushy zone,
compaction, interdendritic spacing
∗ Corresponding author.
Email address: [email protected] (R. Deguen).
Preprint submitted to Elsevier Science
18 January 2008
1
Introduction
As the Earth’s core is gradually cooling down, the inner core is freezing from
the liquid core mixture (Jacobs, 1953) thought to be mostly iron-nickel alloyed
with a small quantity of lighter elements. Although the nature and relative
abundances of these light elements are still controversial (Poirier, 1994a), mineralogical models consistent with the seismological constraints indicate that
their concentration is greater, perhaps by a factor 4, in the liquid core than in
the inner core (Anderson and Ahrens, 1994): like most alloys, the core mixture is fractionating during the process of crystallization, the liquid core being
slowly enriched in light elements.
The release of light elements which results from the gradual solidification of
the core is thought to be a major source of buoyancy for driving the core
convection and the geodynamo (Braginsky, 1963; Gubbins et al., 2004), but
segregation of light solute may also have dramatic consequences on the structure of the solid inner core itself (Fearn et al., 1981). It is well known from
metallurgical experiments that solute segregation during solidification can result in the supercooling of the liquid close to the solidification front. This
supercooling is usually suppressed by either solidification of isolated crystals
in a slurry layer, or by the development of a mushy zone, where solid cells or
dendrites coexist with a solute rich liquid (e.g. Kurz and Fisher, 1989).
Loper and Roberts (1981) and Fearn et al. (1981) show that the conditions at
the inner core boundary (ICB) are almost certainly favorable to the formation
of either a slurry layer or a mushy layer. Fearn et al. (1981) gave preference
to the latter and concluded that a mushy zone of considerable depth, possibly
extending to the inner core center, must have developed at the ICB. More
recently, Shimizu et al. (2005) quantitatively studied the possible regimes of
solidification (slurry layer or mushy zone) and found that a dendritic regime
is more probable than a slurry layer regime, because of the difficulty to supply
enough nuclei to feed a slurry layer. Some seismological study (e.g. Cao and
Romanowicz, 2004) also argue in favor of the existence of a mushy zone at the
top of the inner core.
Morse (1986, 2002) challenged this view arguing that convective motion should
quantitatively remove light solute from the inner core boundary, thus suppressing supercooling. From an analogy between the growth of the inner core
and cumulate solidification in magma chambers, Morse (2002) concluded that
the ICB must be microscopically flat, and rejected the idea of a mushy zone
at the top of the inner core. Cumulates and dendritic layers have dynamics
similar in many respects, both being reactive porous media, but their development occurs through two physically different processes. Cumulates form by
sedimentation of early formed crystals from the melt, the possible porosity
2
resulting from a competition between sedimentation of crystals and solidification in the layer. In contrast, a dendritic layer develops as a consequence of an
instability of an initially plane solid-liquid interface. Whether Morse’s analysis
is appropriate or not to the formation of a dendritic layer may therefore be
questionable, but it is probably correct that advective transport of solute may
affect the solidification regime of the ICB.
The morphological stability of a solidification front has been intensively studied both theoretically and experimentally (e.g. Mullins and Sekerka, 1963;
Kurz and Fisher, 1989; Davis, 2001) but, as noted by Shimizu et al. (2005),
the freezing of the inner core occurs in very unusual conditions: the growth
rate is very small, about 6 orders of magnitude smaller than in typical laboratory experiments, and the pressure gradient has a key role, the newly formed
solid being hotter than the adjacent liquid.
In this work, we will take into account first order effects of advective transport
and investigate possible effects of the very slow solidification rate and of the
pressure gradient on the stability of a solidification front (sections 2, 3 and
4). In section 5, the thermophysical parameters and conditions relative to the
crystallization of the inner core will be evaluated from a survey of the literature
and from the use of thermodynamic constraints. The stability of the ICB will
then be discussed and we shall argue that the existence of a mushy zone is
very likely. Finally, in section 6, we will estimate various length scales relevant
to the structure of a mushy inner core, such as its depth and interdendritic
spacing.
2
Supercooling in the vicinity of an interface
When a dilute alloy is frozen from the melt, the newly formed solid usually
differs in composition from the liquid, in a way which depends on the solubility of the solute in the solid and liquid phases. This solute redistribution is
conveniently described by the distribution coefficient k = c s /c, where c s and
c are respectively the mass fractions of solute in the solid and in the liquid. In
the common case where k < 1, solidification results in the rejection of solute
from the solid phase, the liquid phase being enriched in solute in the vicinity
of the solidification front.
As the melting temperature is lowered by the presence of solute, the liquidus
temperature at the interface is lower than that of the liquid ahead of the
interface, which is richer in solute. If the actual temperature gradient at the
solidification front is smaller than the gradient of the melting temperature,
the temperature of the liquid close to the interface is smaller than the liquidus
temperature, i.e. the liquid is supercooled. This non-equilibrium state usually
3
does not persist and results in the formation of either a mushy zone, through
a destabilization of the solidification front, or a slurry layer.
Denoting T ℓ and Tm the temperature in the liquid and the melting temperature
respectively, the criterion for supercooling is:
dT ℓ dTm <
dz 0
dz 0
(1)
where the subscript 0 denotes the value at the solid-liquid interface, and where
the z axis points toward the liquid. In the core, the melting temperature
depends on both the solute concentration c and the pressure P . Defining the
liquidus and Clapeyron slopes as:
∂T m ∂T m mc =
< 0 and m P =
> 0,
∂c P
∂P c
(2)
the supercooling criterion can be written:
GℓT − m P GP < m c Gc
(3)
dc dP dT ℓ , Gc =
and GP =
are respectively the thermal,
where
=
dz 0
dz 0
dz 0
chemical and pressure gradients at the solid-liquid interface. In the solidification conditions of the core, the pressure gradient is negative and m P positive
(e.g. Boehler, 1993), thus m P GP is negative and the pressure field acts against
supercooling. In contrast, the temperature and concentration fields both promote supercooling, as GℓT < 0 and m c Gc > 0. We can therefore predict that
the pressure field should have a stabilizing effect on the solidification front
whereas temperature and solutal fields should be destabilizing.
GℓT
3
The effect of advective transport on the supercooling criterion
If motion is present in the liquid, advective transport of solute, and, to a
smaller extent, of heat, may affect the solutal and thermal profiles near the
solid-liquid boundary, and thus the degree of supercooling. In this section,
we consider the effects of buoyancy-driven convection on the mean solute and
thermal fields. Let us consider the directional solidification of a dilute binary
alloy at constant velocity V . The solute has a concentration c∞ in the bulk of
the liquid and a chemical diffusivity Dc in the liquid. The coordinate system is
fixed on the moving front, the z axis pointing toward the liquid perpendicularly
to the solidification front. The liquid has a velocity u.
In the melt, the solute concentration c is given by the equation of conservation
4
in the bulk:
∂c
+ (u · ∇)c = Dc ∇2 c,
∂z
with boundary conditions at infinity:
−V
c → c∞
at z → ∞,
(4a)
(4b)
and at the interface z = 0, given by the equation of conservation of solute at
the solidification front:
Gc = (cs0 − c0 )
V
V
= c0 (k − 1) ,
Dc
Dc
(4c)
where cs0 and c0 are the concentration in the solid and liquid phases at the
interface. If advective transport is negligible, a solute boundary layer of thickness δc = Dc /V will build up, and after a transient stage during which the
solute concentration in the solid at the solidification front gradually increases
from kc∞ to c∞ , the system reaches a steady state where the concentration of
solute c(z) in the liquid is
c(z) = c∞ + c∞
1−k
exp (−z/δc )
k
(5)
(e.g. Kurz and Fisher, 1989; Davis, 2001). The concentration in the liquid at
the interface is c0 = c∞ /k and the chemical gradient at the interface is
G c = c∞
k−1 V
.
k Dc
(6)
However, if the solidification is very slow, as it is at the ICB, advection may
affect strongly the concentration gradient at the interface. With a solidification
velocity of 10−11 m.s−1 and a chemical diffusivity of order 10−9 m2 .s−1 , the
chemical diffusive boundary layer thickness is δc = Dc /V ∼ 100 m. Defining
the solutal Rayleigh number Ras as:
Ras =
βgGcδc4
,
νDc
(7)
where β ∼ 1 (Gubbins et al., 2004) is the compositional expansion coefficient,
g = 4.4 m.s−2 is the magnitude of gravity at the ICB (Dziewonski and Anderson, 1981) and ν ∼ 10−6 m2 s−1 is the kinematic viscosity of the liquid,
we found Ras ∼ 1021 . This is much larger than the critical Rayleigh number for the Rayleigh-Taylor instability. The purely diffusive boundary layer
is therefore subject to convective instabilities, which remove light solute from
the vicinity of the interface much more efficiently than diffusion alone.
The purely diffusive concentration profile described by equation (5) is therefore not relevant to the case of the Earth’s inner core solidification, and effects
5
of advective solute transport on the mean concentration profile must be investigated. In the case of purely diffusive solute transport, c0 is equal to c∞ /k.
This is the maximum allowed value as a greater value will imply a solute
concentration in the solid greater than c∞ . If convective motions are present,
advective transport results in the depletion of solute close to the solidification
front, and the concentration in the liquid at the boundary will be closer to c∞
as convection is more vigorous, whereas concentration in the solid will tend to
kc∞ . Because solute conservation implies that the concentration in the solid
cannot exceed c∞ and must be greater than kc∞ , the concentration gradient
Gc at the interface is bounded by:
c∞ (k − 1)
V
k−1 V
≤ G c ≤ c∞
,
Dc
k Dc
(8)
the upper bound being the value of the chemical gradient in the purely diffusive case. This gradient tends toward its lower bound as the efficiency of
advective transport is enhanced; in what follows, we will consider that the
actual chemical gradient can be approximated by its lower bound:
Gc ≃ c∞ (k − 1)
V
,
Dc
(9)
which should be compared with equation (6). If k is small, the convective
chemical gradient can be considerably smaller than the diffusive gradient.
With a no-slip boundary condition, u(x, y, 0) = 0 for all (x, y), equation (4a)
taken at z = 0 gives the second derivative of the mean concentration profile
at the interface:
d2 c V dc 1
=−
(10)
= − Gc .
2
dz 0
Dc dz 0
δc
In what follows, the vertical variations of the mean concentration field in the
vicinity of the interface will be described by a second order Taylor expansion:
c(z) = c0 + Gc z − Gc
1 z2
.
δc 2
(11)
Because the thermal diffusivity is much larger than the solute diffusivity Dc ,
advection affects the solute flow much more strongly than the thermal flow,
and we will ignore the effects of advection on the thermal field, i.e. the mean
temperature is solution of the equation of diffusion in the liquid and in the
solid:
∂T ℓ
= DTℓ ∇2 T ℓ ,
(12)
−V
∂z
∂T s
= DTs ∇2 T s ,
(13)
−V
∂z
where DTℓ and DTs are the thermal diffusivities in the liquid and solid respectively. The thermal gradient at the interface is given by the heat balance at
6
the solid-liquid interface:
Lv V = κs GsT − κℓ GℓT ,
(14)
where Lv is latent heat per unit volume, κs and κℓ are the thermal conductivities in the solid and liquid respectively and GsT and GℓT are the gradient of
temperature at the interface in the solid and liquid respectively. The temperature field can be expanded as
T (z) = T0 + GℓT z − GℓT
1 z2
,
δT 2
(15)
where δT = DT /V .
Shimizu et al. (2005) did not take into account effects of convection on the
thickness of the boundary layer, and assumed it to be given by the diffusive
boundary layer thickness δc = Dc /V . They study the possibility that a fraction
of the solidification occurs in the supercooled zone, whose thickness has been
taken to be the diffusive boundary layer thickness, that is ∼ 100 m. With
this value, they found that it is unlikely that a significant fraction of the
solidification occurs in a slurry layer, because of the difficulty of supplying
continuously enough nuclei. Taking into account advective transport leads
to a much thinner supercooled zone, with smaller supercooling, where the
amount of crystals solidified is probably much smaller than expected from
the model of Shimizu et al. (2005). We will therefore consider that all the
solidification occurs at the solid-liquid interface, and that growth of crystals
in the supercooled zone is negligible.
4
Linear stability analysis of the growth interface of a binary alloy
in a pressure gradient
4.1 Solutal and thermal instabilities
Considerations of section 3 allow us to determine a mean state for the solute
and thermal fields, hereafter noted c̄ and T̄ , respectively described by equations (11) and (15); the pressure field is taken to be hydrostatic, the pressure
gradient being GℓP = −ρℓ g.
In this section we consider the stability of the planar interface against infinitesimal perturbations of the mean fields. The interface is not planar anymore, but has an infinitesimal topography h(x, y, t) which temporal evolution
we study. The solute and thermal fields can be expressed as the sum of the
mean field and infinitesimal disturbances : c(x, y, z, t) = c̄(z) + c̃(x, y, z, t) and
7
T (x, y, z, t) = T̄ (z) + T̃ (x, y, z, t). The linear stability analysis we propose here
is similar in principle to Mullins and Sekerka’s analysis (Mullins and Sekerka,
1963). The two main differences concern the formulation of the basic state,
which is considered here to be altered by convective motions, and the dependence of the melting temperature on pressure.
The thermal and solutal fields must satisfy:
(i) the equations of conservation of solute and heat in the liquid and solid
phases:
∂c
∂c
−V
= Dc ∇2 c,
(16a)
∂t
∂z
∂T ℓ
∂T ℓ
−V
= DTℓ ∇2 T ℓ ,
∂t
∂z
∂T s
∂T s
−V
= DTs ∇2 T s .
∂t
∂z
Diffusion of solute in the solid is neglected.
(16b)
(16c)
(ii) the boundary conditions at the interface. At thermodynamic equilibrium,
the temperature at the interface T I must obey the Gibbs-Thomson relation:
T I = Tm + ΓH + mP GℓP h + mc c,
(16d)
which states that the temperature at the interface is equal to the melting temperature of the curved interface, when variations with pressure, concentration
and interface curvature are taken into account. Tm is the solvent melting temperature of a flat interface at z = 0, H ≃ ∇2 h is the interface curvature and
Γ = Tm γ/Lv is the Gibbs coefficient, where γ is the liquid-solid interfacial
energy.
In addition, the thermal and solutal fields must respectively satisfy the heat
balance and the solute conservation at the interface:
Lv vn (x, y) = κs ∇T s − κℓ ∇T ℓ · n(x, y),
(16e)
(cs − c) vn (x, y) = Dc ∇c · n(x, y),
(16f)
where vn (x, y) ≃ V + ∂h/∂t is the speed of the front and n is the unit normal
vector pointing toward the liquid.
(iii) boundary conditions at infinity: the perturbations must decay to zero at
infinity (but see discussion below).
Equations (16) lead to a system in perturbation quantities h, c̃, T̃ ℓ and T̃ s
which is linearized, allowing to seek solutions for the perturbed fields under
8
the following normal mode form:
h
 
 
 c̃ 
 
 ℓ
T̃ 
 
 
T̃ s




 h1 




 c1 (z) 
 exp(ωt + ikx x + iky y)

T ℓ (z) 

 1


=

(17)
T1s (z)
where ω is the growth rate of the disturbance and kx and ky are the wave numbers along the interface in the direction x and y. From equations (16a,b,c) and
the boundary condition (iii), c1 (z), T1ℓ (z) and T1s (z) are found to be proportional to exp(βz), exp(β ℓ z) and exp(β s z) respectively, where β, β ℓ and β s
are
βs = −
βℓ = −

1 V 
1 −
2 DTs

1 V 
1 +
2 DTℓ
β=−
1V
2 Dc

v
u
u
t
v
u
u
t
1+4
1+4
v
u
u

t
1 + 1 + 4
DTs kh
V
DTℓ kh
V
Dc k h
V
!2
!2
!2









≥ 0,
≤−
≤−
V
,
DTℓ
V
.
Dc
Using expression (17) in the perturbation equations leads to a set of homogeneous, linear equations in h1 , c1 , T1ℓ and T1s which have nontrivial solutions if
the following dispersion equation is satisfied:


ω = αmc Gc − αℓ GℓT − αs GsT + mP GP − Γkh2






,

Dc /V 2
Lv
−
mc Gc ,
 κs β s − κℓ β ℓ
βDc /V + 1 − k
9
(18)
where:
kh =
q
kx2 + ky2 ,
κℓ
κℓ β ℓ + κℓ V /DTℓ
∈
0
,
α =
κℓ β ℓ − κs β s
κℓ + κs
#
"
κs
κs DTℓ
κs β s + κs V /DTs
s
,
∈ ℓ
α =− ℓ ℓ
κ β − κs β s
κ + κs κℓ DTs
β + V /Dc
α=
∈ [0 1].
β + (1 − k)V /Dc
ℓ
"
#
Stability of the interface against infinitesimal perturbations depends on the
sign of ω: the solidification front is stable if ω is negative for all wave numbers
whereas it is unstable if ω is positive for any wave number. As κs β s −κℓ β ℓ ≥ 0,
βDc /V + 1 − k ≤ 0 and mc Gc > 0, the denominator is always positive and so
the sign of ω depends only on the sign of the numerator, which is a combination
of temperature, concentration and pressure gradient weighted by wave number
dependent functions.
Convection has been taken into account when estimating the mean chemical
gradient, but to be rigorous, we should have considered the effects of convection on the perturbed fields as well. We might have imposed, as Coriell et al.
(1976) or Favier and Rouzaud (1983) did, that perturbations of the solute field
must decay to zero on a finite length, imposed by convection, but numerical
calculations not presented here show that this does not significantly affect the
prediction of the stability of the interface. This can be understood as follows.
With the additional hypothesis of a small solutal Peclet number V /kh Dc (that
is, for instability wavelengths small compared to 2πDc /V ≃ 300 m, an hypothesis which will be shown in section 5 to be well verified), β is very close
to −kh , and α is very close to one because kh ≫ V /Dc . If taken into account,
advection of the perturbed solutal field would decrease the decay length of
the perturbations, and so increase β. This would make α to be even closer to
one, but will hardly affect the value of the chemical term in the numerator.
Effects on the stability limit and on the critical wavelength would therefore
be insignificant. However, an increase of β would result in a decrease of the
denominator, and hence in an increase of ω: the instability growth rate may be
underestimated by equation (18), but this point will not weaken our analysis
of the ICB solidification regime.
The dispersion relation confirms the qualitative differences between constant
pressure solidification experiments and the crystallization of the inner core,
as seen from the criterion of supercooling. In contrast with usual solidification, temperature gradients are negative and destabilizing, as α is positive. As
mP GP is negative, the pressure field stabilizes the interface against topography
10
perturbations.
The differences between usual solidification and crystallization in a pressure
field are illustrated in a ln V versus ln c diagram (figure 1) constructed from
equation (18). The dashed curve is the neutral curve for the purely solutal stability problem: temperature gradients are taken to be constants and rejection
of latent heat at the interface is neglected. In this limiting case, the neutral
curve is very similar to that of constant pressure solidification (e.g. Davis,
2001): the curve possesses two asymptotic straight lines of slopes -1 and +1
which correspond to the constitutional supercooling limit and to the absolute
stability limit respectively, where the stabilizing effect of the surface tension
becomes dominant. In constant pressure solidification, allowing perturbations
of the thermal field slightly stabilizes the interface for small c, and flattens
the nose of the marginal curve (Davis, 2001). In the system considered here,
the thermal gradient is destabilizing, and even when solidifying a pure melt
(no solutal destabilization), the solidification front may become unstable if
the solidification velocity is high enough. The neutral curve of the complete
thermo-solutal problem is shown as a solid line in figure 1. For low c∞ , the
instability is mainly thermally driven and the critical solidification velocity
is independent of the concentration. As for the purely solutal destabilization,
the system reaches an absolute stability limit for very high solidification rates,
with the neutral curve being independent of c∞ at small concentration.
Effects of advection on the stability limit is illustrated in figure 2, which represent neutral curves obtained with two different expressions for the chemical
gradient at the interface. The dashed curve has been obtained with the upper
bound of Gc , in inequality (8), and therefore corresponds to a purely diffusive
solutal transport. The solid curve has been obtained with the lower bound of
Gc , which corresponds to the case of maximum advective transport of solute. If
the segregation coefficient k is small, convection has a considerable stabilizing
effect: at a given solute concentration c, the critical solidification velocity can
be as much as an order of magnitude greater than in the case of no convection.
4.2 Damping of instabilities by solid deformation
The only effect of the pressure gradient which has been considered so far is the
pressure dependence of the melting temperature. Yet, because of the difference of density ∆ρ between the solid and the liquid, a surface topography may
induced horizontal pressure gradients which may tend to flatten the interface,
therefore acting against its destabilization. To quantify the possible importance of solid flow on the interface stability, we will estimate the timescale
of isostatic adjustment, assuming that the viscous deformation is driven by a
11
ln V
STABLE
UNSTABLE
STABLE
ln c∞
Fig. 1. Neutral curves for linear morphological stability, constructed from equation
(18). The dashed line is the neutral curve for the purely solutal stability problem.
The solid line is the neutral curve for the thermo-solutal stability problem.
balance between the pressure gradient and the viscous force:
0 = −∇p + η∇2 u,
(19)
where η is the solid state viscosity. Let us assume that the interface has a
topography of amplitude h which varies on a length scale λ. We are still
dealing with linear stability analysis and infinitesimal perturbations, so that,
as λ is finite, h ≪ λ. The finite amplitude case, i.e. dendrites compaction,
will be considered in section 6.1.2. Because h ≪ λ, the deformation induced
by the topography must be accommodated in depth, on a length-scale ∼ λ,
and the horizontal and vertical velocity u and w must be of the same order
of magnitude. The horizontal pressure gradient is of order ∆ρgh/λ, ∇2 u is of
order u/λ2, so that:
∆ρg
hλ.
(20)
u∼w∼
η
The timescale of isostatic adjustment can be defined as the ratio of the topography to the vertical velocity τ = h/w, and is equal to:
τ∼
η
.
∆ρgλ
12
(21)
−9
10
−10
V (m.s−1)
10
UNSTABLE
−11
10
−12
10
−13
10
STABLE
−14
10
−6
10
−5
10
−4
−3
10
10
−2
10
−1
10
0
10
Mass fraction, c∞
Fig. 2. Neutral curves for purely diffusive transport of solute (dashed line) and for
advective transport (solid line). mc = −103 K, k = 0.2.
In the limit of h ≪ λ which is considered here, τ has a finite value even
for an infinitesimal amplitude h of the topography. This process is therefore
relevant when dealing with linear stability analysis. We will however consider
that this effect is negligible compared to the destabilizing effect of the thermal
and compositional gradient, an assumption which will be justified a posteriori
in section 5.2 where τ will be compared to the timescale 1/ω of the growth of
an instability.
5
Morphological stability of the ICB
5.1 Thermo-physical parameters, composition, growth rate
As discussed in section 2, the degree of supercooling and the stability of the
boundary depend strongly on the abundance of solute and on the phase diagram of the crystallizing alloy, i.e., the liquidus slope and the segregation
coefficient. The nature and relative abundances of light elements in the core
are still uncertain (Poirier, 1994a), but recent studies seem to show a prefer13
ence for O, S and Si as major light elements (Ringwood and Hibberson, 1991;
Stixrude et al., 1997; Alfè et al., 2002; Rubie et al., 2004). Very little is known
about the phase diagrams of the candidate alloys and it is not even clear
whether those systems have an eutectic or solid-solution behavior (Williams
and Jeanloz, 1990; Knittle and Jeanloz, 1991; Boehler, 1993; Sherman, 1995;
Boehler, 1996). However, a fundamental constraint on the phase diagram of
the iron-major light elements system is that the outer core is richer in light
solute than the inner core. Major light components must therefore have a segregation coefficient k smaller than one and this necessarily imposes that their
liquidus slopes m c are negative near the Fe-rich end.
The liquidus slope and solute concentration of interest here are that of the
fractionating elements. Elements which do not fractionate during the process
of solidification, such as Ni and perhaps S and Si (Alfè et al., 2002), will not
create chemical heterogeneity and hence will not contribute to supercooling
and radial variations in melting temperature.
Assuming ideal mixing, a crude estimate of the liquidus slope of the core
mixture at ICB pressure and temperature may be provided by the van’t Hoff
relation (Chalmers, 1964). We consider here the effect of alloying Fe with a
single light element, of mole fraction x, on the melting temperature of the
mixture. Chemical equilibrium between two multicomponents phases requires
equality of the chemical potentials of each component in the two phases. In
particular, the chemical potentials of the solvent, here Fe, must be equal in
the liquid and in the solid: µℓFe = µsFe . In an ideal solid or liquid solution, the
chemical potential of a component i is expressed as µi = µ◦i + RT ln xi , where
xi is the mole fraction of component i and µ◦i is the chemical potential of pure
i. Equality of the chemical potentials of iron in the liquid and solid phases
then requires that
µ◦Fes + RT ln xsFe = µ◦Feℓ + RT ln xℓFe ,
(22)
which can be rewritten, using the fact that the mole fraction of the solute is
x = 1 − xFe and that xs = kxℓ , as
µ◦Fes − µ◦Feℓ
1 − xℓ
= R ln
.
T
1 − kxℓ
(23)
Taking the derivative of equation (23) with respect to T , and using the Gibbs14
Helmoltz relation then gives
∂
∂T
1 − xℓ
ln
1 − kxℓ
!
P
∂ µ◦Fes − µ◦Feℓ
,
=
∂T
RT
h◦Fes − h◦Feℓ
=
,
RTm2
MFe L
,
=
RTm2
!
(24)
where h◦Fes and h◦Feℓ are the molar enthalpy of solid and liquid Fe respectively,
L = (h◦Fes − h◦Feℓ )/MFe is the latent heat of pure iron and MFe is the atomic
weight of iron. Assuming k to be constant, for a dilute solution (xℓ ≪ 1), we
obtain
∂
∂T
1 − xℓ
ln
1 − kxℓ
!
P
1
k
=
−
ℓ
1 − kx
1 − xℓ
∂xℓ ∼ (k − 1)
∂T P
!
∂xℓ ∂T P
(25)
which together with (24) yields, as m c can be written as ∂Tm /∂xℓ ,
mc ≃
RTm◦ 2
(k − 1) ≃ (6 ± 3)(k − 1) × 103 K
MFe L
(26)
where Tm◦ is the melting temperature of pure iron at ICB pressure and m c
is given in Kelvin per atomic fraction; values and incertitudes of Tm◦ and L
are from table 1. This relation means that for an ideal solution, the liquidus
slope and the segregation coefficient are related by parameters which are independent of the nature of the alloying element. For a non-partitioning element
(k = 1), the ideal liquidus slope is equal to zero, whereas a highly partitioning
element (k small compared to 1) will have a high liquidus slope. Through this
relation, constraints on k may provides informations on the liquidus slope of
the core mixture.
From estimates of the volume change during melting and of the density jump
at the inner core boundary, Anderson and Ahrens (1994) estimated the ratio
of light elements in the outer core to that in the inner core to be approximately
4 to 1. This gives a global segregation factor k = 0.25 and, by equation (26),
a liquidus slope mc ≃ −4.5 ± 2.5 × 103 K. Note that the segregation factor
estimated here is an effective segregation factor, which is higher than the
thermodynamic one, and should give a lower bound of |mc |. This estimate may
be appropriate if there is only one light element in the core, but finer estimates
are needed if there are several light elements of comparable concentrations.
As an example, the ab initio simulations of Alfè et al. (2002) suggest that
the outer core may be composed of ≃ 10 mole% of S and/or Si and ∼ 8
mole% (≃ 2 wt.%) of O. According to Alfè et al. (2002), S and Si do not
15
significantly fractionate and the density jump at the ICB may be accounted
for by fractionation of oxygen alone, whose segregation coefficient has been
estimated to be 0.02. With this value, mc tends to its k = 0 bound mc ≃
−6 ± 3 × 103 K. In what follows, two chemical models of the core will be
considered: one with a single light element of concentration ≃ 10 wt.%, and
the other with only one fractionating light element (but several light elements),
oxygen, of concentration ≃ 2 wt.%.
Our relation is similar to the one derived by Alfè et al. (2002). Stevenson
(1981) and Anderson and Duba (1997) estimated the melting point depression by assuming equilibrium between a pure solid and an alloyed liquid, and
therefore obtained the upper bound of our estimate (i.e. our k = 0 value).
Those theoretical estimates are in poor agreement with experiments. Experimental results on the melting temperature of the Fe-O system (Knittle and
Jeanloz, 1991; Boehler, 1993) predict a small melting temperature depression,
and perhaps a solid-solution behavior. The case of the Fe-S system if more
controversial. Williams and Jeanloz (1990) results suggest that the eutectic
behavior persists at high pressure, and predict a significant melting point depression (mc is of order −5 × 103 K at core-mantle boundary pressure). In
contrast Boehler (1996) found that the Fe-FeS eutectic melting depression becomes much smaller at high pressures, and conclude that this supports the
possibility of solid-solution between Fe and FeS at core pressures. To our
knowledge, there is no experimental work at this pressure range dealing with
other candidate alloys. In the present work, values of mc between −102 K and
−104 K will be considered.
The interfacial energy of iron at ICB conditions can be deduced from estimates
of the latent heat of crystallization, because those two parameters both derive
from the difference of atoms bonds energy between the solid and liquid phase.
The interfacial energy per atom γa can be calculated to be 1/4 of the atomic
latent heat La for a flat close-packed surface (Chalmers, 1964). Estimates of the
latent heat of iron at ICB conditions range from 600 kJ kg−1 to 1200 kJ kg−1
(Poirier, 1994b; Anderson and Duba, 1997; Laio et al., 2000; Vočadlo et al.,
2003a) and from these values, we estimate the interfacial energy per unit area
γ to be 0.4 ± 0.2 J m−2 , which can be compared to the 0.204 J.m−2 value at
standard conditions (Chalmers, 1964).
Buffett et al. (1992) proposed an analytical model of growth of the inner core
and found
q that the radius increase at leading order as the root square of time,
r = ric t/a, where ric is the present radius of the inner core and a is its age.
The present solidification velocity is then V = ric /2a. If the inner core is young
(e.g. Labrosse et al., 2001; Nimmo et al., 2004), i.e. a ∼ 1 Ga, V is found to be
2 × 10−11 m.s−1 . On the other hand, if the inner core nucleated around 3 Ga
ago, as Christensen and Tilgner (2004) claim, V could be of order 6 × 10−12
m.s−1 . Wen (2006) observed a temporal change of travel time of the PKiKP
16
phase between the two events of an earthquake doublet, indicating a localized
change of the inner core radius of about 1 km in ten years. This observation
may be interpreted as reflecting episodic growth of the inner core, coupled
with non stationary convection in the outer core (Wen, 2006). The resulting
instantaneous solidification velocity is ≃ 10 km/10 years ≃ 10−6 m.s−1 , which
is much higher than the mean solidification velocity estimated from models of
the core thermal history.
Other parameters used in this study, with values currently found in the literature, are listed in table 1.
Table 1
Parameters used in this study.
a
Clapeyron slope
mP
≃ 10−8 K Pa−1
Liquidus slope
mc
−102 to −104 K
Light elements concentration
c∞
2 to 10 wt.%
Growth rate of the inner core
V
6 × 10−12 to 2 × 10−11 m s−1
Temperature at the ICB
Ticb
5000 to 6000 K
Specific heat at constant pressure
cP
860 J kg−1 K−1
Latent heat of crystallization
L
600
Thermal conductivity in the liquid
κℓ
63 W m−1 K−1
d
Thermal conductivity in the solid
κs
79 W m−1 K−1
d
Thermal diffusivity in the liquid
ℓ
DT
6 × 10−6 m2 s−1
d
Thermal diffusivity in the solid
s
DT
7 × 10−6 m2 s−1
d
Chemical diffusivity
Dc
10−9 m2 s−1
Viscosity of the inner core
η
1016 to 1021 Pa s
Liquid-solid interfacial energy
γ
0.4 ± 0.2 J m−2
Gibbs-Thomson coefficient
a from Anderson and Duba (1997).
b see text.
c from Poirier (1994b).
d from Stacey and Anderson (2001).
Γ
≃ 2 × 10−7 K.m
17
c
b
b
a,c
to 1200 kJ kg−1
c
b
b
b
a
b
5.2 Supercooling and stability analysis
The liquid at the ICB is supercooled if condition (3) is satisfied. This criterion
can be rewritten, using the heat balance at the interface (equation (14)), as:
κs s
Lv V
+
m
G
−
G > ρgm P .
c c
κℓ
κℓ T
(27)
Depending on the age of the inner core, on the value of the thermal conductivity and on the hypothetical presence of radioactive elements, the thermal
gradient GsT in the inner core at the ICB may vary widely (Yukutake, 1998).
However, the thermal term Lv V /κℓ , which is the contribution to supercooling
from the heat released by crystallization, is high enough to balance the pressure term if the solidification velocity is higher than ∼ 5 × 10−12 m.s−1 . This
is roughly equal to the lowest estimates of V , and, as the term −GsT is positive, it is thus likely that thermal terms alone are sufficiently high to ensure
supercooling. The chemical term m c Gc is probably much higher, at least one
order of magnitude greater than the pressure term if mc is as small as −102
K and three order of magnitude greater if mc = −104 K, and the conclusion
that the vicinity of the ICB is indeed supercooled seems to be inescapable.
We now use the dispersion equation (18) to investigate more quantitatively
the stability of an initially plane solid-liquid interface in the actual conditions
of the ICB. As discussed in section 3, we choose to take the lower bound of the
chemical gradient, given by equation (9), so that the effect of convection on
the basic state may be overestimated rather than underestimated. We tested
several values of each parameter within their uncertainty ranges and found
that for all plausible sets of values, the ICB is unstable. The liquidus slope is by
far the most critical parameter. Because the chemical gradient is proportional
to mc , the two orders of magnitude uncertainty on mc propagate directly
in the uncertainties on the location of the marginal stability curve. Neutral
curves for linear stability are plotted in figure 3, with mc = −102 K, −103
K and −104 K, GsT = 0 K.m−1 and k = 0.25. Despite the strong dependence
on mc of the neutral curve location, the solidification velocity is two orders
of magnitude greater than the critical velocity for a liquidus slope as small
as −102 K, and three or four orders of magnitude greater than the critical
velocity for a liquidus slope of −103 K or −104 K. If the growth of the inner
core is episodic, with a much higher instantaneous solidification velocity, the
ICB would be even more unstable.
In figure 4, the growth rate of infinitesimal perturbations has been plotted
against wave length, at the conditions of the ICB, for three different values of
mc . The corresponding time-scales range from a few years to about 300 years.
We may now estimate the timescale of isostatic adjustment, given by equation
(21), and compare it to the timescale of instability growth. The solid inner core
18
−9
10
mc=−102 K
−10
mc=−103 K
−1
V (m.s )
10
UNSTABLE
mc=−104 K
−11
10
ICB
−12
10
−13
10
STABLE
−14
10
−6
10
−5
10
−4
−3
10
10
−2
10
−1
10
0
10
Mass Fraction, c
∞
Fig. 3. Neutral curves for linear morphological stability, with mc = −102 K (dotted
line), mc = −103 K (dashed line) and mc = −104 K (solid line). GsT = 0 K.m−1
and k = 0.25. The straight lines cross is the location of the ICB in the stability
diagram. Uncertainties in V result from uncertainties in the age of the inner core.
The value of c∞ depends on the chemical model chosen: c∞ ≃ 10 wt.% if there is
only one dominant light element, and c∞ ≃ 2 wt.% if the model of Alfè et al. (2002)
is adopted.
viscosity is poorly constrained: estimates range from 1016 Pa.s to more than
1021 Pa.s (Buffett, 1997; Yoshida et al., 1996). A lower estimate of the viscous
flow timescale may be given by taken η = 1016 Pa.s. With ∆ρ = 600 kg.m−3 ,
g = 4.4 m.s−1 (Dziewonski and Anderson, 1981), the isostatic adjustment
timescale is of order 1014 s ≃ 3 × 106 years for λ ∼ 1 cm, about four or six
orders of magnitude greater than the timescale of instability growth: isostatic
adjustment will not delay the instability. Our results are consistent with the
conclusion of Shimizu et al. (2005) that the timescale of dendrites growth is
very short compared to the timescale of inner core growth, suggesting that a
mushy layer will indeed form at the ICB.
Results have been presented here for parameters (growth rate, thermal and
pressure gradient) inferred for the present state of the inner core, but when
dealing with the internal structure of the inner core, it is of equal interest
to investigate what was the solidification regime during its past history. As
19
−9
x 10
8
4
m =−10 K
c
7
m =−103 K
c
ω (s−1)
6
2
mc=−10 K
5
4
3
2
1
0
−1
−3
10
10
−2
−1
λ (m)
10
0
10
Fig. 4. Growth rate of infinitesimal perturbations against wave length, at the conditions of the ICB, for mc = −104 K (solid line), −103 K (dashed line) and −102 K
(dash-dotted line). k = 0.02 and Lv = 600 kJ.kg−1 .
explained before, the solidification regime depends primarily upon the thermal, solutal and pressure gradients at the interface, the thermal and solutal
fields being destabilizing whereas the pressure field is stabilizing. Because the
solidification velocity was most certainly greater in the past than it is today,
the rates of release of heat and solute were also greater; thermal and solutal
gradients were therefore steeper (more destabilizing). In addition, the liquidus
slope m P G P is less steep at deeper depth (because the gravity field, and hence
the pressure gradients decrease to zero at the center of the Earth), and therefore less stabilizing. At first order, all terms seem to act in the same way, and
it is then likely that the solidification of the inner core has been dendritic for
most of its history.
20
6
Length scales of the mush
6.1 Vertical length scales
6.1.1 Thermodynamic depth of the mushy zone
In the laboratory, and in metallurgical applications, the mushy zone depth is
typically a few centimeters. However, because of the very small temperature
gradient in the inner core, and of the additional effect of pressure, the temperature in the inner core remains close to the melting temperature, which
suggests that liquid enriched in solute may remain thermodynamically stable
at considerable depths (Fearn et al., 1981).
While freezing occurs in the mushy zone, interdendritic melt is further enriched
in solute, lowering its melting temperature further. An enriched liquid phase
can coexist with the solid phase as long as the actual temperature is above
the melting temperature. The depth δTh at which a liquid of concentration
c = co + ∆c can be in thermodynamic equilibrium with the surrounding solid
phase is then given by equating the actual and melting temperatures:
T (δTh ) = Tm (δTh ).
(28)
To a good approximation, the acceleration of gravity is linear in r in the inner
core, so that pressure is quadratic in radius. Assuming for convenience the
temperature profile to be quadratic as well (which is a not so bad approximation if the cooling rate is approximately constant within the inner core), the
following expression for δTh can be found:
δTh
≃
ric
s
|mc |∆c
,
∆Θ
(29)
where ∆Θ ≃ 150 to 200 K is the difference between the actual temperature and
the melting temperatures (at outer core composition) at the center of the inner
core (Yukutake, 1998). The precise form of the temperature profile (quadratic
or not) is not of great importance for our order of magnitude estimates.
The mushy zone can extend to the center of the inner core (δTh /ric = 1) if the
variation of melting temperature with pressure is compensated by |mc |∆c. The
maximum allowable light elements concentration is the concentration at the
eutectic (if it exists), and the maximum value of |mc |∆c is therefore |mc |(cE −
c0 ). Taking cE = 25 at.% as a plausible value (Stevenson, 1981), we found from
equation (29) that δTh = ric if |mc | is greater than 1.3 × 103 K. On the other
hand, if |mc | is as small as 102 K, the resulting depth of liquid thermodynamic
equilibrium is of the order of 300 km.
21
6.1.2 A collapsing mushy zone?
As noted by Loper (1983), a fundamental observation is that the ICB appears
to be sharp on seismic wavelength scale (∼ 10 km), which means that the
solid fraction at the top of the inner core must become significant within a
few kilometers in depth, a fact that seems at first view hard to reconcile with
the presence of a hundred kilometers thick mushy zone. Loper (1983) argued
that convective motions in the mush may cause the dendrites to thicken and
calculated that the solid fraction may become of order one at ≃ 300 m below
the top of the mushy zone, thus explaining the seismic observations.
In addition to this process, the solid mass fraction may also increase in depth
because of gradual collapse of the dendrites under their own weight. The physical process at stake is quite similar to the isostatic adjustment considered in
section 4: the density difference between the solid and the liquid implies horizontal pressure gradients which increase with depth in the mushy zone, making
the dendrites to broaden at their base by viscous solid flow. The timescale of
dendrites widening may be estimated by considering equilibrium between the
horizontal pressure gradient and the viscous force in the solid (equation (19)),
as was done is section 4. Here the horizontal length scale λ, say the diameter
of a dendrite, is supposed to be small compared to the height h of a dendrite.
As before, the horizontal velocity at depth h is of order:
u∼
∆ρg
hλ,
η
(30)
but the timescale of interest is now the ratio of horizontal length scale to the
horizontal velocity τc = λ/u, so that the timescale of dendrite widening - or
compaction - is:
η
,
(31)
τc ∼
∆ρgh
which is inversely proportional to the height of the dendrite. As long as the
timescale of dendrites growth is small compared to τc , the depth of the mushy
zone increases. When increasing the height of the dendrites, the timescale of
dendrites widening decreases, and a stationary state is eventually reached,
where the growth of the dendrites by solidification at their tips is balanced
by the collapse of the mushy zone. This stationary depth is then found by
equating τc to the timescale of dendrites growth, which is h/V when the system
is stationary. This yields:
s
ηV
h∼
.
(32)
∆ρg
Uncertainties in h come mostly from uncertainties in the viscosity. With η =
1016 Pa.s, h is found to be a few meters, a very small value, whereas η = 1021
Pa.s leads to h ∼ 1 km (Sumita et al., 1996). Although estimates of h span
more than two orders of magnitude, they are in any case significantly smaller
than the thermodynamic estimate of the depth of the mushy zone. This means
22
that as a result of compaction, the solid fraction may become of O(1) at a few h
in depth, well before the thermodynamic limit of liquid equilibrium is reached.
6.2 Horizontal length scale: interdendritic spacing
Interdendritic spacing λ1 (the distance between two dendrites tips) is a length
scale of considerable interest for the structure of the inner core. Because each
columnar crystal is usually made of many dendrites (Kurz and Fisher, 1989),
the interdendritic spacing gives a lower bound of the grain size, and hence
may gives hints on the deformation mechanism and viscosity relevant to the
possible viscous deformation of the inner core. In addition, the primary dendrite spacing is needed for estimating the permeability of the mush, which
is roughly proportional to λ21 (Bergman and Fearn, 1994). Permeability is a
necessary parameter for the study of the mush hydrodynamics (e.g. Bergman
and Fearn, 1994; Le Bars and Worster, 2006), or for quantitative compaction
models (Sumita et al., 1996).
We make the assumption that dendrites have an axisymmetric shape described
by a function g(z), which is linked to the solid fraction f (z) by the relation:
f (z) =
πg 2 (z)
,
A(λ1 )
(33)
where A(λ1 ) is the horizontal surface area occupied by a dendrite. In a cubic
dendritic array, A(λ1 ) is simply λ21 , whereas in a hexagonal dendritic array,
A(λ1 ) = 3/2 tan(π/3)λ21 ≃ 0.86λ21 .
f and g 2 are proportional through a constant involving λ1 , so that if f is
known, an additional constraint on g is enough to determine λ1 . This constraint can be given by Langer and Müller-Krumbhaar’s theory of dendrite
tip radius selection (Langer and Müller-Krumbhaar, 1977), in which the dendrite tip radius R is equal to the shortest wavelength λi for which the interface
is unstable. This theory is in very good agreement with experiments, and we
will follow it in the present work. It can be shown (see Kurz and Fisher, 1989)
that at low solutal Peclet number (Pec = V R/D ≪ 1), a condition well satisfied here, the chemical gradient Gc at the dendrite tip is equal to the lower
bound of the chemical gradient we used in our stability analysis (equation (9)).
Estimates of λi from our stability analysis thus directly give the adequate R.
A good approximation for R at low solutal and thermal Peclets numbers can
be derived from equation (18), noting that the chemical term in the numerator
of (18) is much bigger than the thermal and pressure terms (Kurz and Fisher,
1989):
s
ΓD
R ≃ 2π
(34)
mc c∞ (k − 1)V
23
R ranges from ∼ 1 mm if mc = −104 K to ∼ 1 cm if mc is as low as −102 K
(see figure 4). Uncertainties from other parameters than mc are much smaller.
On the other hand, the dendrite tip radius R is equal to the radius of curvature
of g taken in z = 0 which is by definition equal to:
R=−
(1 + g ′ (0)2 )3/2
g ′′ (0)
(35)
where R is defined to be positive. Inserting equation (33) into (35), with the
boundary condition f (0) = 0, yields
A(λ1 ) =
or, for an hexagonal dendritic array:
λ1 =
v
u
u
u
u
t
2πR
.
∂f ∂z ICB
2πR
.
∂f 0.86
∂z ICB
(36)
(37)
Although simple, this relation is very general, the only assumptions made here
being that of stationary state and that of an axisymmetric dendrite tip. If the
appropriate assumptions on solute transport and selection of dendrites tip
radius are made, classical models of primary dendrite spacing (Trivedi, 1984;
Hunt, 1979) follow from equation (37).
These models have been derived for non-convecting mushy zones where the
solid fraction increase smoothly from zero to one at the thermodynamic depth
of the mush, that is, ∂f /∂z|ICB scales as 1/δTh 1 . However, seismology tells
us that the ICB is sharp on a scale of 10 km, which means that the scale
length of solid fraction increase at the top of the inner core is obviously not
the thermodynamic mushy zone depth, and must be at maximum ∼ 10 km.
Therefore a minimum order of magnitude of ∂f /∂z|ICB is 1/10 km = 10−4
m−1 . This may give upper bounds for λ1 ; we found λ1 ∼ 10 m if mc = −104
K and λ1 ∼ 30 m if mc = −102 K. Tighter constraints may be found if we
could estimate more precisely ∂f /∂z|ICB . Compaction is not expected to affect
significantly the dendrites spacing because it is not effective in the very top
of the mush. The solid fraction profile in the upper few meters, from which
the primary spacing results, is more probably controlled by convection. In his
1
Note that, in laboratory experiments, with no pressure gradient and with a
constant temperature gradient δTh ∼ mc (ce − c∞ )/GT , so that, as R ∝ V −1/2 ,
−1/2
λ1 ∝ GT V −1/4 . This scaling law has been shown to be in very good agreement
with experiments (e.g. Kurz and Fisher, 1989).
24
convective mushy zone model, Loper (1983) found that ∂f /∂z|ICB ∼ 3 × 10−3
m−1 . With this value and our estimates of R, we found λ1 to be 1 to 5 meters.
In Loper’s estimate, ∂f /∂z|ICB is proportional to the typical velocity W of
descending liquid close to the ICB, taken to be around 10−6 m.s−1 . This value
is quite uncertain, but as λ1 is inversely proportional to the square root of
∂f /∂z|ICB , a change of two orders of magnitude in W is needed in order to
change the order of magnitude of λ1 . As a consequence, and considering all
sources of uncertainties, it appears difficult to have an interdendritic spacing
of more than a few tens of meters, our preferred estimate being a few meters. If
the growth of the inner core is episodic, the interdendritic spacing is expected
to be smaller. As λ1 ∝ V −1/4 , taking V = 10−6 m.s−1 instead of V = 10−11
m.s−1 would result in an interdendritic spacing about twenty times smaller
than estimated above, that is, a few tens of centimeters.
Our estimates are significantly smaller than estimate of a few hundred meters
from scaling laws (Bergman, 1998). This is not surprising because those scaling laws have been derived for non-convecting (and non-compacting) mushy
zones, where f is linearly increasing in the whole thermodynamic mushy zone.
Assuming for a demonstrative purpose that f increases linearly from 0 to 1
within the thermodynamic mushy zone, whose depth is taken equal to the inner core radius, equation (37) gives λ1 ∼ 100 − 300 m, in very good agreement
with Bergman’s estimate.
Our assumption of an axisymmetric dendritic shape is justified if the iron
phase at inner core conditions is fcc or bcc iron (Vočadlo et al., 2003b), but
might not hold if it is ǫ-iron, as hcp materials (e.g. ice or zinc) usually have
plateshaped dendrites, known as platelets (Bergman et al., 2003). If iron dendrites are indeed platelets, estimating the inter-platelets spacing would require
some modifications of our analysis to take into account the specific geometry
of platelets. Although this is expected to give quantitatively different results,
this should not alter the qualitative conclusion that convection in the mush
reduces the interdendritic spacing, and that interdendritic spacing at ICB
should be much smaller than suggested by classic scaling laws.
7
Conclusion
The morphological stability of an initially plane solidification front at the ICB
conditions has been investigated. Despite the stabilizing effects of convection
and of the pressure gradient, a continuous solidification implies non-zero solutal and thermal gradients at the ICB which, for plausible parameters values,
are high enough for the interface to be destabilized. Because the conditions in
the past were even more destabilizing, it is probable that the ICB has been
dendritic through most of the inner core history.
25
Thermodynamic considerations predict a very thick mushy zone which, as
noted by Fearn et al. (1981), could possibly extend to the center of the Earth.
However, considerable uncertainties on the phase diagram do not allow a precise estimate of the thermodynamic depth, which may be only a few tens of
kilometers if the liquidus slope is small. The most superficial part of the inner
core may be understood as a collapsing mushy zone, where both convection
and compaction act to rapidly increase the solid fraction within a length scale
probably smaller than 1 km, making the ICB to appear seismically sharp.
The length scale of the thermodynamic depth of the mushy zone and the
compaction length scale, although clearly different, are unconstrained by the
current knowledge of the phase diagram of the core mixture and of the solid
iron viscosity at inner core conditions. Progress in the determination of these
parameters may greatly help the understanding of the inner core structure.
We tried to constrain the primary dendrite spacing λ1 of the mushy zone. λ1
appears to depend on the vertical derivative of the solid fraction at the top
of the mush which, in turn, depends on the vigor of convection in the mush.
Here again, it is difficult to make precise and reliable estimates, but we found
that the interdendritic spacing is most probably smaller than a few tens of
meters, and possibly only a few meters.
Whether or not a significant amount of melt may subsists at large depth remains an open question. If the mush permeability is of order λ21 /100 (Bergman
and Fearn, 1994), our estimate of λ1 suggests permeability values higher than
10−2 m2 . The work of Sumita et al. (1996) suggests that with such a high
permeability, the liquid will be very efficiently removed from the inner core
by compaction, and that the residual liquid fraction will be essentially zero.
A high permeability in the mush does not however rule out the possibility
that unconnected, trapped liquid pockets persist in depth. The answer to this
question may depend in part on surface tension driven processes. Sintering,
i.e. migration of grain boundaries driven by surface tension, may play an important role in the redistribution of the liquid phase and have consequences
on the efficiency of compaction, in a way which will depend on the wetting
properties of the melt. If the (unknown) dihedral angle of light elements rich
liquid iron in contact with solid iron is greater than 60◦ (Bulau et al., 1979),
the liquid phase is expected to become unconnected at a given liquid fraction,
thus living a residual liquid phase in the inner core.
Acknowledgements
We thank the two anonymous reviewers for their careful reviews and constructive suggestions. We are grateful to Thierry Duffar, Hisayochi Shimizu, JeanLouis Le Mouël and Jean-Paul Poirier for helpful discussions. Careful read26
ing and useful comments by Dominique Jault, Alexandre Fournier, Philippe
Cardin, Franck Plunian and Elisabeth Canet were much appreciated.
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